In the adjoining figure, PA ⊥ AB, QB ⊥ AB and PA=QB. If PQ intersects AB at O, show that O is the midpoint of AB as well as that of PQ.

Question

In the adjoining figure, PA &#x22A5; AB, QB &#x22A5; AB and PA=QB. If PQ intersects AB at O, show that O is the midpoint of AB as well as that of PQ.

Philoid · Accepted Answer

Given: PA ⊥ AB, QB ⊥ AB and PA=QB

To prove: AO = OB and PO = OQ

It is given that PA ⊥ AB and QB ⊥ AB.

This means that ∆PAO and ∆QBO are right angled triangles.

It is also given that PA=QB

Now in ∆PAO and ∆QBO,

∠OAP = ∠OBQ = 90°

PO = OQ

Hence by hypotenuse-leg congruency,

∆PAO ≅ ∆QBO

∴AO = OB and PO = OQ ….by cpct

Hence proved that AO = OB and PO = OQ